3.710 \(\int (d+e x)^m \left (a+c x^2\right )^3 \, dx\)
Optimal. Leaf size=223 \[ -\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac{\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]
[Out]
((c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*c*d*(c*d^2 + a*e^2)^2*(
d + e*x)^(2 + m))/(e^7*(2 + m)) + (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*
x)^(3 + m))/(e^7*(3 + m)) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(4 + m))/(e^7
*(4 + m)) + (3*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (6*c^3*d
*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + m))
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Rubi [A] time = 0.324233, antiderivative size = 223, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059
\[ -\frac{4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac{3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac{\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac{6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac{6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(a + c*x^2)^3,x]
[Out]
((c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (6*c*d*(c*d^2 + a*e^2)^2*(
d + e*x)^(2 + m))/(e^7*(2 + m)) + (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*
x)^(3 + m))/(e^7*(3 + m)) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(4 + m))/(e^7
*(4 + m)) + (3*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (6*c^3*d
*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + m))
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Rubi in Sympy [A] time = 64.8848, size = 207, normalized size = 0.93 \[ - \frac{6 c^{3} d \left (d + e x\right )^{m + 6}}{e^{7} \left (m + 6\right )} + \frac{c^{3} \left (d + e x\right )^{m + 7}}{e^{7} \left (m + 7\right )} - \frac{4 c^{2} d \left (d + e x\right )^{m + 4} \left (3 a e^{2} + 5 c d^{2}\right )}{e^{7} \left (m + 4\right )} + \frac{3 c^{2} \left (d + e x\right )^{m + 5} \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \left (m + 5\right )} - \frac{6 c d \left (d + e x\right )^{m + 2} \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (m + 2\right )} + \frac{3 c \left (d + e x\right )^{m + 3} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (a e^{2} + c d^{2}\right )^{3}}{e^{7} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(c*x**2+a)**3,x)
[Out]
-6*c**3*d*(d + e*x)**(m + 6)/(e**7*(m + 6)) + c**3*(d + e*x)**(m + 7)/(e**7*(m +
7)) - 4*c**2*d*(d + e*x)**(m + 4)*(3*a*e**2 + 5*c*d**2)/(e**7*(m + 4)) + 3*c**2
*(d + e*x)**(m + 5)*(a*e**2 + 5*c*d**2)/(e**7*(m + 5)) - 6*c*d*(d + e*x)**(m + 2
)*(a*e**2 + c*d**2)**2/(e**7*(m + 2)) + 3*c*(d + e*x)**(m + 3)*(a*e**2 + c*d**2)
*(a*e**2 + 5*c*d**2)/(e**7*(m + 3)) + (d + e*x)**(m + 1)*(a*e**2 + c*d**2)**3/(e
**7*(m + 1))
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Mathematica [A] time = 0.447624, size = 398, normalized size = 1.78 \[ \frac{(d+e x)^{m+1} \left (a^3 e^6 \left (m^6+27 m^5+295 m^4+1665 m^3+5104 m^2+8028 m+5040\right )+3 a^2 c e^4 \left (m^4+22 m^3+179 m^2+638 m+840\right ) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+3 a c^2 e^2 \left (m^2+13 m+42\right ) \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )+c^3 \left (720 d^6-720 d^5 e (m+1) x+360 d^4 e^2 \left (m^2+3 m+2\right ) x^2-120 d^3 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+30 d^2 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-6 d e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right )\right )}{e^7 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m*(a + c*x^2)^3,x]
[Out]
((d + e*x)^(1 + m)*(a^3*e^6*(5040 + 8028*m + 5104*m^2 + 1665*m^3 + 295*m^4 + 27*
m^5 + m^6) + 3*a^2*c*e^4*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)*(2*d^2 - 2*d*e*(
1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 3*a*c^2*e^2*(42 + 13*m + m^2)*(24*d^4 - 24
*d^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6 + 11*m + 6*m^2 +
m^3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4) + c^3*(720*d^6 - 720*d^5
*e*(1 + m)*x + 360*d^4*e^2*(2 + 3*m + m^2)*x^2 - 120*d^3*e^3*(6 + 11*m + 6*m^2 +
m^3)*x^3 + 30*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*d*e^5*(120 +
274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2 + 73
5*m^3 + 175*m^4 + 21*m^5 + m^6)*x^6)))/(e^7*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 +
m)*(6 + m)*(7 + m))
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Maple [B] time = 0.017, size = 1140, normalized size = 5.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(c*x^2+a)^3,x)
[Out]
(e*x+d)^(1+m)*(c^3*e^6*m^6*x^6+21*c^3*e^6*m^5*x^6+3*a*c^2*e^6*m^6*x^4-6*c^3*d*e^
5*m^5*x^5+175*c^3*e^6*m^4*x^6+69*a*c^2*e^6*m^5*x^4-90*c^3*d*e^5*m^4*x^5+735*c^3*
e^6*m^3*x^6+3*a^2*c*e^6*m^6*x^2-12*a*c^2*d*e^5*m^5*x^3+621*a*c^2*e^6*m^4*x^4+30*
c^3*d^2*e^4*m^4*x^4-510*c^3*d*e^5*m^3*x^5+1624*c^3*e^6*m^2*x^6+75*a^2*c*e^6*m^5*
x^2-228*a*c^2*d*e^5*m^4*x^3+2775*a*c^2*e^6*m^3*x^4+300*c^3*d^2*e^4*m^3*x^4-1350*
c^3*d*e^5*m^2*x^5+1764*c^3*e^6*m*x^6+a^3*e^6*m^6-6*a^2*c*d*e^5*m^5*x+741*a^2*c*e
^6*m^4*x^2+36*a*c^2*d^2*e^4*m^4*x^2-1572*a*c^2*d*e^5*m^3*x^3+6432*a*c^2*e^6*m^2*
x^4-120*c^3*d^3*e^3*m^3*x^3+1050*c^3*d^2*e^4*m^2*x^4-1644*c^3*d*e^5*m*x^5+720*c^
3*e^6*x^6+27*a^3*e^6*m^5-138*a^2*c*d*e^5*m^4*x+3657*a^2*c*e^6*m^3*x^2+576*a*c^2*
d^2*e^4*m^3*x^2-4812*a*c^2*d*e^5*m^2*x^3+7236*a*c^2*e^6*m*x^4-720*c^3*d^3*e^3*m^
2*x^3+1500*c^3*d^2*e^4*m*x^4-720*c^3*d*e^5*x^5+295*a^3*e^6*m^4+6*a^2*c*d^2*e^4*m
^4-1206*a^2*c*d*e^5*m^3*x+9336*a^2*c*e^6*m^2*x^2-72*a*c^2*d^3*e^3*m^3*x+2988*a*c
^2*d^2*e^4*m^2*x^2-6480*a*c^2*d*e^5*m*x^3+3024*a*c^2*e^6*x^4+360*c^3*d^4*e^2*m^2
*x^2-1320*c^3*d^3*e^3*m*x^3+720*c^3*d^2*e^4*x^4+1665*a^3*e^6*m^3+132*a^2*c*d^2*e
^4*m^3-4902*a^2*c*d*e^5*m^2*x+11388*a^2*c*e^6*m*x^2-1008*a*c^2*d^3*e^3*m^2*x+547
2*a*c^2*d^2*e^4*m*x^2-3024*a*c^2*d*e^5*x^3+1080*c^3*d^4*e^2*m*x^2-720*c^3*d^3*e^
3*x^3+5104*a^3*e^6*m^2+1074*a^2*c*d^2*e^4*m^2-8868*a^2*c*d*e^5*m*x+5040*a^2*c*e^
6*x^2+72*a*c^2*d^4*e^2*m^2-3960*a*c^2*d^3*e^3*m*x+3024*a*c^2*d^2*e^4*x^2-720*c^3
*d^5*e*m*x+720*c^3*d^4*e^2*x^2+8028*a^3*e^6*m+3828*a^2*c*d^2*e^4*m-5040*a^2*c*d*
e^5*x+936*a*c^2*d^4*e^2*m-3024*a*c^2*d^3*e^3*x-720*c^3*d^5*e*x+5040*a^3*e^6+5040
*a^2*c*d^2*e^4+3024*a*c^2*d^4*e^2+720*c^3*d^6)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+
6769*m^3+13132*m^2+13068*m+5040)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.250423, size = 1688, normalized size = 7.57 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a^3*d*e^6*m^6 + 27*a^3*d*e^6*m^5 + 720*c^3*d^7 + 3024*a*c^2*d^5*e^2 + 5040*a^2*
c*d^3*e^4 + 5040*a^3*d*e^6 + (c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 7
35*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7)*x^7 + (c^3*d*e
^6*m^6 + 15*c^3*d*e^6*m^5 + 85*c^3*d*e^6*m^4 + 225*c^3*d*e^6*m^3 + 274*c^3*d*e^6
*m^2 + 120*c^3*d*e^6*m)*x^6 + 3*(a*c^2*e^7*m^6 + 1008*a*c^2*e^7 - (2*c^3*d^2*e^5
- 23*a*c^2*e^7)*m^5 - (20*c^3*d^2*e^5 - 207*a*c^2*e^7)*m^4 - 5*(14*c^3*d^2*e^5
- 185*a*c^2*e^7)*m^3 - 4*(25*c^3*d^2*e^5 - 536*a*c^2*e^7)*m^2 - 12*(4*c^3*d^2*e^
5 - 201*a*c^2*e^7)*m)*x^5 + (6*a^2*c*d^3*e^4 + 295*a^3*d*e^6)*m^4 + 3*(a*c^2*d*e
^6*m^6 + 19*a*c^2*d*e^6*m^5 + (10*c^3*d^3*e^4 + 131*a*c^2*d*e^6)*m^4 + (60*c^3*d
^3*e^4 + 401*a*c^2*d*e^6)*m^3 + 10*(11*c^3*d^3*e^4 + 54*a*c^2*d*e^6)*m^2 + 12*(5
*c^3*d^3*e^4 + 21*a*c^2*d*e^6)*m)*x^4 + 3*(44*a^2*c*d^3*e^4 + 555*a^3*d*e^6)*m^3
+ 3*(a^2*c*e^7*m^6 + 1680*a^2*c*e^7 - (4*a*c^2*d^2*e^5 - 25*a^2*c*e^7)*m^5 - (6
4*a*c^2*d^2*e^5 - 247*a^2*c*e^7)*m^4 - (40*c^3*d^4*e^3 + 332*a*c^2*d^2*e^5 - 121
9*a^2*c*e^7)*m^3 - 8*(15*c^3*d^4*e^3 + 76*a*c^2*d^2*e^5 - 389*a^2*c*e^7)*m^2 - 4
*(20*c^3*d^4*e^3 + 84*a*c^2*d^2*e^5 - 949*a^2*c*e^7)*m)*x^3 + 2*(36*a*c^2*d^5*e^
2 + 537*a^2*c*d^3*e^4 + 2552*a^3*d*e^6)*m^2 + 3*(a^2*c*d*e^6*m^6 + 23*a^2*c*d*e^
6*m^5 + 3*(4*a*c^2*d^3*e^4 + 67*a^2*c*d*e^6)*m^4 + (168*a*c^2*d^3*e^4 + 817*a^2*
c*d*e^6)*m^3 + 2*(60*c^3*d^5*e^2 + 330*a*c^2*d^3*e^4 + 739*a^2*c*d*e^6)*m^2 + 24
*(5*c^3*d^5*e^2 + 21*a*c^2*d^3*e^4 + 35*a^2*c*d*e^6)*m)*x^2 + 12*(78*a*c^2*d^5*e
^2 + 319*a^2*c*d^3*e^4 + 669*a^3*d*e^6)*m + (a^3*e^7*m^6 + 5040*a^3*e^7 - 3*(2*a
^2*c*d^2*e^5 - 9*a^3*e^7)*m^5 - (132*a^2*c*d^2*e^5 - 295*a^3*e^7)*m^4 - 3*(24*a*
c^2*d^4*e^3 + 358*a^2*c*d^2*e^5 - 555*a^3*e^7)*m^3 - 4*(234*a*c^2*d^4*e^3 + 957*
a^2*c*d^2*e^5 - 1276*a^3*e^7)*m^2 - 36*(20*c^3*d^6*e + 84*a*c^2*d^4*e^3 + 140*a^
2*c*d^2*e^5 - 223*a^3*e^7)*m)*x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5
+ 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m*(c*x**2+a)**3,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.219677, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(e*x + d)^m,x, algorithm="giac")
[Out]
Done